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Compound Poisson process

A compound Poisson process is a continuous-time stochastic process defined as S(t) = \sum_{i=1}^{N(t)} X_i, where \{N(t)\}_{t \geq 0} is a Poisson process with rate \lambda > 0, and the X_i are independent and identically distributed random variables independent of N(t), representing the random sizes or "jumps" associated with each Poisson event.^[1]^[2] This construction extends the standard Poisson process—where each jump size is fixed at 1—by incorporating variability in the magnitude of events, making it suitable for modeling cumulative effects like totals over time.^[1]^[3] Key properties of the compound Poisson process include stationary and independent increments, meaning the distribution of S(t + h) - S(t) depends only on h and is independent of the history up to time t.^[1] Its characteristic function is given by \phi_{S(t)}(\theta) = \exp\left( \lambda t \left( \mathbb{E}[e^{i \theta X_1}] - 1 \right) \right), which reflects its infinitely divisible nature and connection to the compound Poisson distribution for the increments.^[1]^[4] The mean and variance of S(t) are \lambda t \mathbb{E}[X_1] and \lambda t \mathbb{E}[X_1^2], respectively, allowing for analytical computation in many cases.^[3] In applications, compound Poisson processes are widely used in actuarial science and risk theory to model aggregate claims in insurance, where N(t) counts the number of claims and X_i represents individual claim severities.^[1]^[3] They also appear in queueing theory for total service times, finance for modeling jumps in asset prices, and other fields like reliability engineering or population dynamics to capture random accumulations of events. As a Lévy process with finite activity, it serves as a foundational model in stochastic calculus and limit theorems for sums of random variables.^[4]

Definition and Construction

Formal Definition

The compound Poisson process is a continuous-time stochastic process \{X(t)\}_{t \geq 0} defined by

X(t) = \sum_{i=1}^{N(t)} Y_i,

where N(t) denotes a Poisson process with positive intensity parameter \lambda > 0, and \{Y_i\}_{i=1}^\infty is a sequence of independent and identically distributed real-valued random variables with common cumulative distribution function F_Y, independent of the counting process \{N(t)\}_{t \geq 0}.^[5]^[2]^[6] The jump sizes Y_i may be positive, negative, or of mixed sign, and there is no inherent restriction to finite variance or higher moments unless explicitly assumed in a given context.^[5]^[2] The process starts at the origin, satisfying X(0) = 0 almost surely, as the empty sum is taken when N(0) = 0.^[5]^[6] A discrete-time analogue for intuition is the compound binomial process, where the number of summands follows a binomial distribution over a fixed number of trials instead of a Poisson counting process.^[7]

Components and Construction

A compound Poisson process is built upon two primary components: a homogeneous Poisson process and a sequence of independent and identically distributed (i.i.d.) jump sizes. The Poisson process, denoted \{N(t), t \geq 0\}, serves as the counting mechanism for the occurrence of jumps. It is a right-continuous, non-decreasing stochastic process with N(0) = 0 and independent, stationary increments, characterized by a constant intensity rate \lambda > 0.^[8] The number of events N(t) in the interval [0, t] follows a Poisson distribution with parameter \lambda t, meaning P(N(t) = n) = e^{-\lambda t} (\lambda t)^n / n! for n = 0, 1, 2, \dots.^[9] The interarrival times between successive events in the Poisson process are independent exponential random variables with rate \lambda, hence mean $1/\lambda. The arrival times are constructed as partial sums T_n = \sum_{i=1}^n \tau_i, where each \tau_i is an interarrival time, resulting in T_n following a gamma distribution with shape parameter n and rate \lambda. This structure ensures that events occur at random times with no fixed pattern, but at an average rate of \lambda per unit time.^[8]^[9] The jump sizes, denoted \{Y_i\}_{i=1}^\infty, form a sequence of i.i.d. real-valued random variables independent of the Poisson process, drawn from an arbitrary probability distribution with cumulative distribution function F. If the moments exist, the jump sizes have mean \mu_Y = \mathbb{E}[Y_1] and variance \sigma_Y^2 = \mathrm{Var}(Y_1). Common examples include exponential distributions for positive jumps, as in modeling claim sizes in insurance, or normal distributions for symmetric perturbations around zero.^[8]^[9] The compound Poisson process X(t) is then formed by superposing these components: X(t) = \sum_{i=1}^{N(t)} Y_i, which sums the jump sizes corresponding to all events up to time t; if N(t) = 0, then X(t) = 0. Jumps occur precisely at the arrival times T_i \leq t, accumulating the value Y_i at each such point. The resulting sample paths are piecewise constant, holding steady between arrival times and undergoing discontinuous jumps of size Y_i at T_i. These paths are right-continuous with left limits (càdlàg), ensuring well-defined limits from the left at jump points and continuity from the right.^[8]^[9]^[10]

Key Properties

Moment-Generating and Characteristic Functions

The moment-generating function of a compound Poisson process X(t) provides a key analytical tool for deriving moments and understanding its distributional properties. Let N(t) be a Poisson process with rate \lambda > 0, and let Y_1, Y_2, \dots be independent and identically distributed random variables, independent of N(t), with common moment-generating function M_Y(s) = \mathbb{E}[e^{s Y_1}] for s in the domain where it exists. Then X(t) = \sum_{i=1}^{N(t)} Y_i (with the empty sum equal to 0 when N(t) = 0), and the moment-generating function of X(t) is

M_{X(t)}(s) = \exp\left( \lambda t (M_Y(s) - 1) \right).

^[11] This formula is derived by conditioning on the value of N(t). Specifically,

\mathbb{E}[e^{s X(t)} \mid N(t) = n] = \left( M_Y(s) \right)^n,

since the conditional distribution of X(t) given N(t) = n is the distribution of the sum of n i.i.d. copies of Y_1. Averaging over the Poisson distribution of N(t) with mean \lambda t then yields

M_{X(t)}(s) = \sum_{n=0}^\infty \left( M_Y(s) \right)^n \frac{(\lambda t)^n e^{-\lambda t}}{n!} = e^{-\lambda t} \exp\left( \lambda t M_Y(s) \right) = \exp\left( \lambda t (M_Y(s) - 1) \right).

This derivation relies on the independence of the increments and the jump sizes.^[11] Analogously, the characteristic function of X(t) is obtained by replacing the moment-generating function with its Fourier analog. Let \phi_Y(u) = \mathbb{E}[e^{i u Y_1}] denote the characteristic function of Y_1, for u \in \mathbb{R}. Then

\phi_{X(t)}(u) = \exp\left( \lambda t (\phi_Y(u) - 1) \right).

The derivation follows identically via conditioning, with the characteristic function of the sum of n i.i.d. Y_i being [\phi_Y(u)]^n, and the Poisson probabilities ensuring the exponential form.^[11] These transform expressions highlight the infinite divisibility of the compound Poisson distribution. For any positive integer n, the characteristic function raised to the power $1/n is \exp\left( (\lambda t / n) (\phi_Y(u) - 1) \right), which is itself the characteristic function of a compound Poisson random variable with rate \lambda t / n and the same jump distribution; thus, the distribution of X(t) can be expressed as the n-fold convolution of identically distributed random variables.^[11] A partial converse due to Feller states that any infinitely divisible distribution on the non-negative integers is compound Poisson.^[11] The exponential form of the characteristic function also implies stability under convolution for independent compound Poisson processes with the same jump distribution. If X_1(t) and X_2(t) are independent compound Poisson processes with rates \lambda_1 and \lambda_2, and common jump characteristic function \phi_Y(u), then the characteristic function of their sum is \exp\left( \lambda_1 t (\phi_Y(u) - 1) \right) \exp\left( \lambda_2 t (\phi_Y(u) - 1) \right) = \exp\left( (\lambda_1 + \lambda_2) t (\phi_Y(u) - 1) \right), which is the characteristic function of a compound Poisson process with rate \lambda_1 + \lambda_2 and the same jumps.^[12]

Stationary and Independent Increments

The compound Poisson process exhibits the property of independent increments, meaning that for any disjoint time intervals [s, t] and [u, v] with [s, t] \cap [u, v] = \emptyset, the increments X(t) - X(s) and X(v) - X(u) are independent random variables.^[13]^[1] This follows directly from the underlying Poisson counting process N(t), which has independent increments, and the i.i.d. jump sizes that are independent of the arrival times.^[13]^[14] Additionally, the process possesses stationary increments, so that the distribution of the increment X(t + h) - X(t) depends only on the length h of the interval and not on the starting time t; in particular, X(t + h) - X(t) \stackrel{d}{=} X(h).^[1]^[14] This stationarity is inherited from the homogeneous Poisson process N(t) with constant rate \lambda, where the number of jumps in any interval of length h follows a Poisson distribution with mean \lambda h, regardless of location.^[13] To sketch the proof, consider the construction X(t) = \sum_{i=1}^{N(t)} J_i, where N(t) is the Poisson process and the J_i are i.i.d. jump sizes independent of N(t). For disjoint intervals, the increments in N(t) are independent Poisson random variables, and the associated jumps are distinct i.i.d. samples, ensuring the overall increments X(t) - X(s) are independent.^[1]^[14] Stationarity holds because the distribution of jumps in any interval of length h is a compound Poisson sum with exactly the same Poisson number of terms and identical jump distribution, independent of the interval's position.^[13]^[1] These increment properties imply that the compound Poisson process is Markovian, as the future evolution X(t + h) - X(t) is independent of the history up to time t and depends only on the current state X(t).^[14] Furthermore, they endow the process with a semigroup structure, where the transition operators P_h f(x) = \mathbb{E}[f(x + (X(t + h) - X(t)) ) \mid X(t) = x] satisfy the semigroup property P_{s+t} = P_s P_t for s, t \geq 0.^[14]

Applications and Examples

In Risk Theory and Insurance

In risk theory, the compound Poisson process serves as a foundational model for aggregate insurance claims, representing the total claims amount S(t) = \sum_{i=1}^{N(t)} Y_i up to time t, where N(t) is a homogeneous Poisson process with intensity \lambda > 0 counting the number of claims, and the Y_i (often denoted C_i) are independent and identically distributed positive random variables denoting individual claim sizes, independent of N(t).^[16] This setup captures the randomness in both claim frequency and severity, assuming claims arrive sporadically and independently with exponentially distributed interarrival times.^[16] The Cramér-Lundberg model, also known as the classical risk process, builds on this by describing the insurer's surplus as U(t) = u + c t - S(t), where u \geq 0 is the initial capital and c > 0 is the constant premium income rate per unit time, assumed to satisfy the net profit condition c > \lambda \mathbb{E}[Y_1] for long-term solvency.^[16] Ruin occurs if the surplus drops below zero at any time, and the ultimate ruin probability is defined as \psi(u) = \mathbb{P}\left( \inf_{t \geq 0} U(t) < 0 \mid U(0) = u \right).^[16] This probability decreases with increasing initial capital u and is central to assessing insurer solvency and required reserves.^[16] A key tool for bounding \psi(u) is the adjustment coefficient R > 0, the unique positive solution to the equation \lambda (M_Y(r) - 1) = c r, where M_Y(r) = \mathbb{E}[e^{r Y_1}] is the moment-generating function of the claim size distribution, assuming it exists in a neighborhood of zero.^[16] Lundberg's inequality then provides an exponential upper bound: \psi(u) \leq e^{-R u}, which quantifies the decay rate of ruin risk and is sharp asymptotically for large u.^[16] The adjustment coefficient can be found using the moment-generating function of the claim sizes.^[17] For a numerical illustration with Poisson claims and exponential claim sizes, consider \lambda = 1 (claims per unit time), claim sizes Y_i \sim \exp(\alpha = 1) (mean 1), and premium rate c = 1.5 (implying a safety loading of 50%). The adjustment coefficient solves to R = 1 - 1/1.5 = 1/3 \approx 0.333, yielding the bound \psi(u) \leq e^{-u/3}.^[17] In this case, the exact ruin probability is \psi(u) = (1/1.5) e^{-u/3} = (2/3) e^{-u/3}, which for u = 3 gives approximately 0.245, compared to the bound of about 0.368.^[17]

In Queueing and Reliability

In queueing theory, the compound Poisson process models the aggregate workload introduced by customer arrivals in the M/G/1 queue, where arrivals follow a Poisson process with rate \lambda > 0 and each customer contributes an independent service time Y_i drawn from a general distribution G with finite mean $1/\mu > 0. The cumulative input up to time t is given by the compound Poisson process X(t) = \sum_{i=1}^{N(t)} Y_i, where N(t) counts the Poisson arrivals. The workload process V(t), which tracks the remaining service time in the system, evolves as the reflection of the net input X(t) - \rho t at zero, where \rho = \lambda / \mu < 1 ensures stability; this structure leverages the independent increments of the compound Poisson input to analyze system congestion.^[18] Analysis of busy periods and waiting times in the M/G/1 queue often relies on an embedded Markov chain observed at customer departure epochs. The chain \{Q_n\}, where Q_n is the number of customers left behind by the nth departing customer, satisfies the transition probabilities P(Q_{n+1} = k \mid Q_n = j) = \int_0^\infty e^{-\lambda u} \frac{(\lambda u)^k}{k!} dG^{(j+1)}(u) for k \geq 0, reflecting the Poisson arrivals during the service of j+1 customers. The stationary distribution of this chain yields the limiting queue length, from which the waiting time distribution follows via the renewal reward theorem, and the busy period—initiated by an arrival to an idle system and ending at the next idle state—has a Laplace transform solving a functional equation derived from the compound Poisson arrivals.^[19]^[18] In reliability analysis, compound Poisson processes underpin shot noise models for cumulative damage or stress from random shocks, particularly with fading effects. The shot noise process is defined as

S(t) = \sum_{i=1}^{N(t)} Y_i e^{-\alpha (t - T_i)},

where N(t) is a Poisson process with rate \lambda, \{T_i\} are the shock arrival times, Y_i > 0 are i.i.d. shock magnitudes independent of N(t), and \alpha > 0 governs the exponential decay, capturing how past shocks diminish in impact over time. This formulation models system degradation where failure occurs if S(t) exceeds a threshold, incorporating the independent increment property of the underlying compound Poisson for tractable moment calculations and tail probabilities. A practical example arises in dam storage models, where inflows follow a compound Poisson process representing stochastic water arrivals—Poisson events with random volumes Y_i—while outflows are linear or state-dependent, such as proportional to current storage level. The storage content Z(t) satisfies dZ(t) = dX(t) - r(Z(t)) dt, reflected at boundaries to prevent negative levels, enabling analysis of overflow risks and depletion probabilities under steady-state conditions. Seminal work derives the stationary distribution and first-exit times using backward equations tailored to the compound Poisson input, informing reservoir operations and flood control strategies.^[20]

Extensions and Relations

As a Lévy Process

A Lévy process is a stochastic process \{X_t\}_{t \geq 0} with càdlàg sample paths, starting at X_0 = 0 almost surely, having stationary and independent increments, and satisfying stochastic continuity, meaning that for every t \geq 0 and \epsilon > 0, \lim_{s \to t} P(|X_t - X_s| > \epsilon) = 0.^[21] The compound Poisson process qualifies as a Lévy process and specifically as a pure-jump process with finite jump activity, where the jump rate is given by the Poisson intensity parameter \lambda > 0.^[22] In this setting, the process exhibits jumps at random times governed by a Poisson process of rate \lambda, with no continuous component or drift in its basic form.^[21] The Lévy measure \nu for a compound Poisson process is defined as \nu(dy) = \lambda F_Y(dy), where F_Y is the probability distribution function of the jump sizes Y, and it is concentrated solely on the jump component with finite total mass \nu(\mathbb{R}^d) = \lambda < \infty.^[21]^[22] This measure fully characterizes the jump structure, reflecting the finite activity nature of the process. In the Lévy-Khintchine triplet representation (b, c, \nu), where b is the drift vector, c is the Gaussian covariance matrix, and \nu is the Lévy measure, the compound Poisson process corresponds to the triplet (0, 0, \nu) with no drift (b = 0) and no diffusion (c = 0).^[21]^[23]

Generalizations to Other Lévy Processes

The compound Poisson process serves as a foundational example within the class of Lévy processes due to its finite jump activity, but generalizations extend this framework by incorporating infinite jump activity, particularly through modifications to the Lévy measure that allow for an infinite mass near zero. In such extensions, the Lévy measure \nu(dx) of the compound Poisson process, which has finite total mass \nu(\mathbb{R} \setminus \{0\}) < \infty, is generalized to forms where \int_{|x|<1} \nu(dx) = \infty, leading to infinitely many small jumps over any finite time interval while maintaining the compensator for the jump component. This transition captures more realistic behaviors in financial modeling and risk assessment, where small fluctuations accumulate continuously.^[24] A prominent generalization is the tempered stable process, where the Lévy measure takes the form \nu(dx) = \left( \alpha^+ x^{-1-\beta^+} e^{-\lambda^+ x} \mathbf{1}_{(0,\infty)}(x) + \alpha^- |x|^{-1-\beta^-} e^{-\lambda^- |x|} \mathbf{1}_{(-\infty,0)}(x) \right) dx for parameters \alpha^\pm > 0, \beta^\pm \in \mathbb{R}, and \lambda^\pm > 0. When \beta^+ , \beta^- < 0, the total mass is finite, recovering a compound Poisson process; however, for \beta^+ , \beta^- \in [0,1), the measure has infinite mass near zero, resulting in infinite small jumps and finite variation paths. Similarly, the variance gamma process arises as a bilateral gamma process or as the limit of a sequence of compound Poisson processes with gamma-distributed jumps, where the intensity increases and jump sizes decrease, yielding infinite activity while preserving infinite divisibility. These processes model heavy-tailed distributions with tempered tails, contrasting the lighter tails of pure compound Poisson models.^[24]^[25] In the subordinator case, where paths are non-decreasing, the compound Poisson process with positive jump sizes Y_i > 0 naturally generalizes to broader subordinators by relaxing the finite activity constraint. Specifically, a compound Poisson subordinator X(t) = \sum_{i=1}^{N(t)} Y_i, with Poisson rate \lambda > 0 and positive i.i.d. jumps, has Lévy measure \nu(dy) = \lambda F(dy) supported on (0,\infty), and adding a non-negative drift Ct ensures monotonicity. Extending to infinite activity subordinators, such as stable subordinators with index \alpha \in (0,1), involves Lévy measures like \nu(dy) = c y^{-1-\alpha} dy for y > 0, leading to infinitely many positive jumps that accumulate to produce strictly increasing paths with infinite variation. These generalizations are crucial for time-changing other processes, like subordinating Brownian motion to generate variance gamma paths.^[22] Further broadening occurs by incorporating a diffusion component, as in the Brownian motion with compound Poisson jumps, where the process becomes X(t) = \mu t + \sigma W(t) + \sum_{i=1}^{N(t)} Y_i, with W(t) a standard Brownian motion. This adds continuous paths to the discontinuous jumps, resulting in a mixed Lévy process with both finite activity jumps and Gaussian diffusion, often used to model asset prices with sudden shocks amid ongoing volatility. Infinite activity variants, like those combining tempered stable jumps with Brownian motion, enhance path irregularity by overlaying infinite small jumps on the continuous component. A key distinction in these generalizations lies in path regularity: finite-activity processes like the compound Poisson exhibit cadlag paths with only finitely many discontinuities over any finite interval, yielding piecewise constant or linear segments interrupted by isolated jumps, and thus finite variation if no diffusion is present. In contrast, infinite-activity Lévy processes, such as stable or tempered stable ones, feature infinitely many jumps in every interval, producing highly irregular paths that are nowhere differentiable and often of infinite variation, resembling fractal-like trajectories despite remaining cadlag. This difference affects simulation and approximation methods, with finite-activity cases allowing exact compound Poisson representations, while infinite-activity requires truncation of small jumps.

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